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Hereditarily non uniformly perfect non-autonomous Julia sets
1. | Department of Mathematics, University of Rhode Island, 5 Lippitt Road, Room 102F, Kingston, RI 02881, USA |
2. | Department of Mathematical Sciences, Ball State University, Muncie, IN 47306, USA |
3. | Course of Mathematical Science, Department of Human Coexistence, Graduate School of Human and Environmental Studies, Kyoto University, Yoshida-nihonmatsu-cho, Sakyo-ku, Kyoto 606-8501, Japan |
Hereditarily non uniformly perfect (HNUP) sets were introduced by Stankewitz, Sugawa, and Sumi in [
References:
[1] |
L. V. Ahlfors, Complex Analysis, McGraw-Hill Book Co., New York, third edition, 1978. An introduction to the theory of analytic functions of one complex variable, International Series in Pure and Applied Mathematics. |
[2] |
Francisco Balibrea,
On problems of topological dynamics in non-autonomous discrete systems, Appl. Math. Nonlinear Sci., 1 (2016), 391-404.
doi: 10.21042/AMNS.2016.2.00034. |
[3] |
E. Camouzis and G. Ladas, Dynamics of Third-Order Rational Difference Equations with Open Problems and Conjectures, Chapman and Hall/CRC, Boca Raton, FL, 2008. |
[4] |
L. Carleson and T. W. Gamelin, Complex Dynamics, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993.
doi: 10.1007/978-1-4612-4364-9. |
[5] |
M. Comerford, A survey of results in random iteration, Proceedings Symposia in Pure Mathematics, American Mathematical Society, 72 (2004), 435–476. |
[6] |
M. Comerford,
Hyperbolic non-autonomous Julia sets, Ergodic Theory Dynamical Systems, 26 (2006), 353-377.
doi: 10.1017/S0143385705000441. |
[7] |
A. Eremenko, Julia Sets are Uniformly Perfect, Preprint, Purdue University, 1992. Google Scholar |
[8] |
K. Falconer, Fractal Geometry, John Wiley & Sons, Ltd., Chichester, third edition, 2014. Mathematical foundations and applications. |
[9] |
J. E. Fornæss and N. Sibony,
Random iterations of rational functions, Ergodic Theory Dynam. Systems, 11 (1991), 687-708.
doi: 10.1017/S0143385700006428. |
[10] |
A. Hinkkanen,
Julia sets of rational functions are uniformly perfect, Math. Proc. Cambridge Philos. Soc., 113 (1993), 543-559.
doi: 10.1017/S0305004100076192. |
[11] |
S. Kolyada and L. Snoha,
Topological entropy of nonautonomous dynamical systems, Random Comput. Dynam., (4) (1996), 205-233.
|
[12] |
R. Mañé and L. F. da Rocha,
Julia sets are uniformly perfect, Proc. Amer. Math. Soc., 116 (1992), 251-257.
doi: 10.1090/S0002-9939-1992-1106180-2. |
[13] |
C. T. McMullen, Complex Dynamics and Renormalization, Volume 135 of Annals of Mathematics Studies, Princeton University Press, Princeton, NJ, 1994.
![]() |
[14] |
C. T. McMullen,
Winning sets, quasiconformal maps and Diophantine approximation, Geom. Funct. Anal., 20 (2010), 726-740.
doi: 10.1007/s00039-010-0078-3. |
[15] |
L. Rempe-Gillen and M. Urbański,
Non-autonomous conformal iterated function systems and Moran-set constructions, Trans. Amer. Math. Soc., 368 (2016), 1979-2017.
doi: 10.1090/tran/6490. |
[16] |
O. Sester,
Hyperbolicité des polynȏmes fibrés, (French) [Hyperbolicity of fibered polynomials], Bull. Soc. Math. France, 127 (1999), 398-428.
|
[17] |
R. Stankewitz,
Uniformly perfect sets, rational semigroups, Kleinian groups and IFS's, Proc. Amer. Math. Soc., 128 (2000), 2569-2575.
doi: 10.1090/S0002-9939-00-05313-2. |
[18] |
R. Stankewitz,
Density of repelling fixed points in the Julia set of a rational or entire semigroup, Ⅱ, Discrete Contin. Dyn. Syst., 32 (2012), 2583-2589.
doi: 10.3934/dcds.2012.32.2583. |
[19] |
R. Stankewitz, H. Sumi and T. Sugawa,
Hereditarily non uniformly perfect sets, Discrete Contin. Dyn. Syst S, 12 (2019), 2391-2402.
doi: 10.3934/dcdss.2019150. |
[20] |
H. Sumi,
Skew product maps related to finitely generated rational semigroups, Nonlinearity, 13 (2000), 995-1019.
doi: 10.1088/0951-7715/13/4/302. |
[21] |
H. Sumi,
Dynamics of sub-hyperbolic and semi-hyperbolic rational semigroups and skew products, Ergodic Theory Dynam. Systems, 21 (2001), 563-603.
doi: 10.1017/S0143385701001286. |
[22] |
H. Sumi,
Semi-hyperbolic fibered rational maps and rational semigroups, Ergodic Theory Dynam. Systems, 26 (2006), 893-922.
doi: 10.1017/S0143385705000532. |
[23] |
H. Sumi,
Dynamics of postcritically bounded polynomial semigroups Ⅲ: classification of semi-hyperbolic semigroups and random Julia sets which are Jordan curves but not quasicircles, Ergodic Theory Dynam. Systems, 30 (2010), 1869-1902.
doi: 10.1017/S0143385709000923. |
[24] |
W. Zhiying,
Moran sets and Moran classes, Chinese Sci. Bull., 46 (2001), 1849-1856.
doi: 10.1007/BF02901155. |
show all references
References:
[1] |
L. V. Ahlfors, Complex Analysis, McGraw-Hill Book Co., New York, third edition, 1978. An introduction to the theory of analytic functions of one complex variable, International Series in Pure and Applied Mathematics. |
[2] |
Francisco Balibrea,
On problems of topological dynamics in non-autonomous discrete systems, Appl. Math. Nonlinear Sci., 1 (2016), 391-404.
doi: 10.21042/AMNS.2016.2.00034. |
[3] |
E. Camouzis and G. Ladas, Dynamics of Third-Order Rational Difference Equations with Open Problems and Conjectures, Chapman and Hall/CRC, Boca Raton, FL, 2008. |
[4] |
L. Carleson and T. W. Gamelin, Complex Dynamics, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993.
doi: 10.1007/978-1-4612-4364-9. |
[5] |
M. Comerford, A survey of results in random iteration, Proceedings Symposia in Pure Mathematics, American Mathematical Society, 72 (2004), 435–476. |
[6] |
M. Comerford,
Hyperbolic non-autonomous Julia sets, Ergodic Theory Dynamical Systems, 26 (2006), 353-377.
doi: 10.1017/S0143385705000441. |
[7] |
A. Eremenko, Julia Sets are Uniformly Perfect, Preprint, Purdue University, 1992. Google Scholar |
[8] |
K. Falconer, Fractal Geometry, John Wiley & Sons, Ltd., Chichester, third edition, 2014. Mathematical foundations and applications. |
[9] |
J. E. Fornæss and N. Sibony,
Random iterations of rational functions, Ergodic Theory Dynam. Systems, 11 (1991), 687-708.
doi: 10.1017/S0143385700006428. |
[10] |
A. Hinkkanen,
Julia sets of rational functions are uniformly perfect, Math. Proc. Cambridge Philos. Soc., 113 (1993), 543-559.
doi: 10.1017/S0305004100076192. |
[11] |
S. Kolyada and L. Snoha,
Topological entropy of nonautonomous dynamical systems, Random Comput. Dynam., (4) (1996), 205-233.
|
[12] |
R. Mañé and L. F. da Rocha,
Julia sets are uniformly perfect, Proc. Amer. Math. Soc., 116 (1992), 251-257.
doi: 10.1090/S0002-9939-1992-1106180-2. |
[13] |
C. T. McMullen, Complex Dynamics and Renormalization, Volume 135 of Annals of Mathematics Studies, Princeton University Press, Princeton, NJ, 1994.
![]() |
[14] |
C. T. McMullen,
Winning sets, quasiconformal maps and Diophantine approximation, Geom. Funct. Anal., 20 (2010), 726-740.
doi: 10.1007/s00039-010-0078-3. |
[15] |
L. Rempe-Gillen and M. Urbański,
Non-autonomous conformal iterated function systems and Moran-set constructions, Trans. Amer. Math. Soc., 368 (2016), 1979-2017.
doi: 10.1090/tran/6490. |
[16] |
O. Sester,
Hyperbolicité des polynȏmes fibrés, (French) [Hyperbolicity of fibered polynomials], Bull. Soc. Math. France, 127 (1999), 398-428.
|
[17] |
R. Stankewitz,
Uniformly perfect sets, rational semigroups, Kleinian groups and IFS's, Proc. Amer. Math. Soc., 128 (2000), 2569-2575.
doi: 10.1090/S0002-9939-00-05313-2. |
[18] |
R. Stankewitz,
Density of repelling fixed points in the Julia set of a rational or entire semigroup, Ⅱ, Discrete Contin. Dyn. Syst., 32 (2012), 2583-2589.
doi: 10.3934/dcds.2012.32.2583. |
[19] |
R. Stankewitz, H. Sumi and T. Sugawa,
Hereditarily non uniformly perfect sets, Discrete Contin. Dyn. Syst S, 12 (2019), 2391-2402.
doi: 10.3934/dcdss.2019150. |
[20] |
H. Sumi,
Skew product maps related to finitely generated rational semigroups, Nonlinearity, 13 (2000), 995-1019.
doi: 10.1088/0951-7715/13/4/302. |
[21] |
H. Sumi,
Dynamics of sub-hyperbolic and semi-hyperbolic rational semigroups and skew products, Ergodic Theory Dynam. Systems, 21 (2001), 563-603.
doi: 10.1017/S0143385701001286. |
[22] |
H. Sumi,
Semi-hyperbolic fibered rational maps and rational semigroups, Ergodic Theory Dynam. Systems, 26 (2006), 893-922.
doi: 10.1017/S0143385705000532. |
[23] |
H. Sumi,
Dynamics of postcritically bounded polynomial semigroups Ⅲ: classification of semi-hyperbolic semigroups and random Julia sets which are Jordan curves but not quasicircles, Ergodic Theory Dynam. Systems, 30 (2010), 1869-1902.
doi: 10.1017/S0143385709000923. |
[24] |
W. Zhiying,
Moran sets and Moran classes, Chinese Sci. Bull., 46 (2001), 1849-1856.
doi: 10.1007/BF02901155. |


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